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Av(123, 231, 1432, 2143)

Permutation examples

length 2: 12, 21

length 3: 132, 213, 312, 321

length 4: 3214, 4132, 4213, 4312, 4321

length 5: 43215, 53214, 54132, 54213, 54312, 54321

ATRAP tree (size 7, depth 4)

Legend

$\mathcal{A}$ = Av(123, 231, 1432, 2143)

$\mathcal{B}$ = Av(12, 321)

$\mathcal{C}$ = Av(12)

$\mathcal{D}$ = Av(12, 21)

Generating function

$A(x) = \frac{- x^{4} + x^{3} + x^{2} - x + 1}{x^{2} - 2 x + 1}$

Coefficients

1, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, ...

System of equations

$\operatorname{F_{10}}{\left (x \right )} = \operatorname{F_{4}}{\left (x \right )} + \operatorname{F_{9}}{\left (x \right )}$

$\operatorname{F_{4}}{\left (x \right )} = 1$

$\operatorname{F_{9}}{\left (x \right )} = \operatorname{F_{5}}{\left (x \right )} + \operatorname{F_{8}}{\left (x \right )}$

$\operatorname{F_{5}}{\left (x \right )} = - \frac{x}{x - 1}$

$\operatorname{F_{8}}{\left (x \right )} = \operatorname{F_{6}}{\left (x \right )} + \operatorname{F_{7}}{\left (x \right )}$

$\operatorname{F_{6}}{\left (x \right )} = \frac{x^{2}}{\left(x - 1\right)^{2}}$

$\operatorname{F_{7}}{\left (x \right )} = - \frac{x^{3}}{x - 1}$